THEORY
Measurement of electrical power
Instantaneous, mean, active, reactive and apparent electrical power, power factor, etc.
We would like to remind you about these basic parameters in electronics and about three-phase measurement
methods.
Definition of electrical power
proportional to the product in the case of electronic wattmeters; this current
or this voltage is then applied to an analogue or digital display.
At a given moment, when a current i travels from generator G to receiver R
in the direction defined by the voltage v delivered by the generator (figure 1),
the instantaneous power supplied to the receiver R is equal to product v.i.
■ The apparent power S = Veff . Ieff, in VA (volt-amperes), defining the voltage
Veff not to be exceeded (insulator breakdown, increase in core loss) and
the intensity Ieff circulating in the receivers.
i
G
Generator
V
The existence of a phase shift ϕ between the current and the voltage leads,
for AC currents, to the introduction of 3 additional quantities:
R
Receiver
figure 1
If the voltage and current are DC, the mean power V. I is equal to the
instantaneous power v.i.
If the voltage and current are sinusoidal AC, there is generally a phase shift ϕ
between the voltage and the current (figure 2).
The instantaneous values of voltage v and current i have the form:
v = Vmax cos ωt
i = Imax cos (ωt - ϕ)
Where ω, the pulse, is proportional to the frequency F (ω = 2πF).
■ The power factor:
P
P
cos ϕ = =
S Veff . Ieff
when the current and voltage are sinusoidal quantities.
■ The reactive power Q = Veff . Ieff . sin ϕ, in rva (reactive volt-amperes).
The latter may be directly measured by a wattmeter if for voltage
V max . cos ωt we substitute a phase-shifted voltage of π/2, i.e.
Vmax x cos (ωt - π/2).
The mean product measured will be
Vmax . Imax . cos (ωt - π/2) x cos (ωt - ϕ) which is expressed by:
Vmax.Imax
Q=
cos (π/2 - ϕ) = Veff . Ieff . sin ϕ
√2
Knowing P and Q, we can calculate the apparent power and the power factor:
V max
Apparent power: S = √P2+Q2
I max
Power factor: PF = P/S = P/√P2+Q2
Knowing the parameters defined above: active power, reactive power, apparent
power, power factor, is fundamental in electrical engineering and enables
accurate calculation of the characteristics of the equipment used: yield, load,
cos ϕ, utilisation limits. The wattmeters used for these measurements are
classified in three major families: electrodynamic, ferrodynamic and electronic.
figure 2
Measurement of active power
The phase shift ϕ is, conventionally, counted as positive when the current is
delayed in relation to the voltage.
The instantaneous power has a value of: Vmax . Imax . cos ω . cos (ωt - ϕ).
You must take the average value of this product during a period to obtain
the expression of the power provided by generator G to receiver R. This power
is called the active power and is expressed by the formula:
Vmax.Imax
P=
cos ϕ = Veff . Ieff . cos ϕ
√2
The wattmeters provide the expression of this product, either by causing
a deviation of the pointer in the case of a device with an electrodynamic
or ferrodynamic moving coil, or by supplying a DC current or a voltage
4
4-wire balanced three-phase measurement (3 phases + neutral)
The intensities circulating in the three phases are equal in terms of rms
values I1 = I2 = I3 and show the same phase shift ϕ in relation to the respective voltages of the 3 phases.
If U1N is the simple voltage measured between phase 1 and neutral, power
P1 supplied by phase 1 will be obtained by connecting a wattmeter as shown
in figure 3.
Its value will be: P1 = U1N. I1.cos ϕ
The total power supplied P will be equal to 3 P1.
CONTACT No. 16
THEORY
➝ ➝ ➝ ➝ ➝ ➝
P = (U1N – U3N. I1 + (U2N – U3N). I2
1
I1
I’
I*
➝ ➝ ➝ ➝
so P = U13. I1 + U23. I2
P1
U’
U*
2 I2
3
and the measurement of the total power may be carried out using two
wattmeters (figure 5).
USE
U 1N
I3
U13 and U23 are the phase-to-phase voltages measured respectively between
phase 1 and phase 3 and then between phase 2 and phase 3.
N
figure 3
Note: The expression P1 = U1N. I1 . cos ϕ in the scalar product of the 2 vectors
➝
➝
U1N and I1 which enables use of the notation
➝ ➝
P1 = U1N. I1
and in three-phase:
➝➝ ➝ ➝ ➝ ➝
P = U1N. I1 + U2N. I2 + U3N. I3
1
2
I1
I*
I’
P1
U* U’
I2
I*
I’
P2
U’
USE
U*
3
I3
Measurement in 3-wire balanced three-phase (3 phases no neutral)
The intensities circulating in the three phases are equal I1 = I2 = I3.
An artificial neutral is created using three resistors R, R et R’. The sum
R’ + r must be equal to R (r is the resistance of the voltage circuit of the unit).
figure 5
Two cases may arise:
a) P1 ≥ 0 and P2 ≥ 0, then Ptotal = P1 + P2
This returns us to the previous case with U 1N between phase 1 and
the artificial neutral (figure 4).
1
I1
I2
2
3
I*
P1
U*
I’
If it is a digital wattmeter we will add together the algebraic values displayed.
U'
USE
Note: it is possible to use a single wattmeter successively connected to 2
positions, using an inverter switch. This type of switch contains auxiliary
contacts ensuring short-circuiting of the unused contacts.
I3
U 1N
R’
b) one wattmeter deviates to the right and the other is as far as it will go to
the left. To read the second; transfer the feed wires to the voltage circuit:
U*.U’ becomes U’.U*.
The value will be considered negative and we will obtain: Ptotal = P1 – P2
R
R
Measurement in 4-wire balanced three-phase (3 phases + neutral)
N
figure 4
1
P1 = Power supplied on phase 1
2
I1
I2
With many wattmeters, the balanced three-phase measurements (3 phases
no neutral) are performed directly; the artificial neutral point recreated by
the resistors R, R and R’ is included in the instrument (astatic wattmeter, CdA
778 wattmeter, for example). This design is shown in the diagram by the
dotted section.
Measurement in 3-wire unbalanced three-phase (3 phases no neutral) method using two wattmeters.
Whether the circuit is balanced or not in the absence of a neutral, there
remains I1 + I2 + I3 = 0.
In this case, the general expression of the power given above is simplified
No. 16 CONTACT
U1N
I’
P1
U*
U’
I*
I’
P2
Totale P supplied = 3 U1N. I1. cos ϕ = 3P1.
3
I*
I3
U2N
USE
U* U’
I*
U3N
P3
I’
U* U’
N
figure 6
We obtain Ptotal = P1 + P2 + P3 (figure 6).
In this case, we must use 3 wattmeters and add the readings together.
If the measurement is stable, we can successively carry out 3 measurements with a single wattmeter. Caution: it is recommended to use a system
preventing the intensity circuits from being cut off during switching.
5